Let's analyze the problem step-by-step.
Problem Summary
- Andrew and Benjamin together can paint a car in 20 hours.
- They start working together, but after 4 hours, Andrew leaves.
- Benjamin finishes the remaining work alone in 24 hours.
- We need to find how long Andrew alone would take to paint the entire car.
Step 1: Define variables
- Let Andrew's rate be AAA cars per hour.
- Let Benjamin's rate be BBB cars per hour.
From the problem:
- Together, they paint 1 car in 20 hours, so their combined rate is:
A+B=120(cars per hour)A+B=\frac{1}{20}\quad \text{(cars per hour)}A+B=201(cars per hour)
Step 2: Work done in the first 4 hours together
In 4 hours, working together, they paint:
4×(A+B)=4×120=420=15of the car4\times (A+B)=4\times \frac{1}{20}=\frac{4}{20}=\frac{1}{5}\quad \text{of the car}4×(A+B)=4×201=204=51of the car
Step 3: Remaining work
The remaining work after 4 hours is:
1−15=451-\frac{1}{5}=\frac{4}{5}1−51=54
Step 4: Benjamin finishes the remaining work alone in 24 hours
Benjamin's rate BBB is:
B=worktime=4524=45×24=4120=130cars per hourB=\frac{\text{work}}{\text{time}}=\frac{\frac{4}{5}}{24}=\frac{4}{5\times 24}=\frac{4}{120}=\frac{1}{30}\quad \text{cars per hour}B=timework=2454=5×244=1204=301cars per hour
Step 5: Find Andrew's rate
Recall:
A+B=120A+B=\frac{1}{20}A+B=201
Substitute B=130B=\frac{1}{30}B=301:
A+130=120A+\frac{1}{30}=\frac{1}{20}A+301=201
A=120−130=360−260=160A=\frac{1}{20}-\frac{1}{30}=\frac{3}{60}-\frac{2}{60}=\frac{1}{60}A=201−301=603−602=601
Step 6: Find how long Andrew alone takes to paint the car
Andrew's rate is 160\frac{1}{60}601 cars per hour, so time taken is:
Time=1A=60 hours\text{Time}=\frac{1}{A}=60\text{ hours}Time=A1=60 hours
Final answer:
Andrew alone can paint the entire car in 60 hours.
